KUALA LUMPUR, Malaysia—In its heyday, Malaysia Airlines was the toast of this steamy Southeast Asian capital.

It flew far-flung routes from Argentina to Croatia to South Africa, even though Malaysia was a developing country. Its $3.5 billion home airport, opened in 1998, was codesigned by a Japanese architect to look like a modernist masterpiece in the jungle, with natural rain forest between terminals. Employees wore uniforms...